49 lines
2.7 KiB
Markdown
49 lines
2.7 KiB
Markdown
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---
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title: Quadratic Equations
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---
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## Quadratic Equations
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_A quadratic equation_ is a polynomial function of degree 2, equated to 0 or a constant.
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The parent equation for a quadratic function is **ax^2+bx+c=0** where x is variable and a, b and c are real constants.
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* 'a' determines how wide or narrow the function is.
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* If |a| is greater than 1, the parabola will be narrow.
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* If |a| is less than 1, the parabola will be wider.
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* Roots of any function are the values of the parameter(s) where the function equates to zero. Roots of a quadratic equation (function actually) are the value of variable (here it is 'x' since the equation we have taken is quadratic in 'x') which satisfies the equation for a given set of constants (here--> a,b,c).
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* Every quadratic equations **ax^2+bx+c=0** can be expressed as **(x-p)(x-q)=0** where p and q will be the roots of the given quadratic equation. These roots may or may not be real in nature.
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* Quadratic functions create a parabola, also known as a 'u' shape.
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* The vertex of a quadratic funtion is the turning point of which the graph reflects itself (hence the vertex also relating to 'axis of symmetry', the line in which a quadratic function reflects).
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* The values of x where the graph of **y=ax^2+bx+c** touches the x axis are the roots of the quadratic equation **ax^2+bx+c=0**.
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## ROOTS
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A quadratic always have 2 roots. In case the quadratic function represents a perfect square, it is said that both the roots have the same value (saying that there is only one root will be wrong since a quadratic equation has to have 2 roots). The nature and value of roots can be calculated using the set of constants associated with it.
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### Nature of Roots
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As stated earlier, the roots of a quadratic equation are not always real. The nature of the roots can be determined easily by calculating the value of D which is given by b^2-4ac
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**D=b^2-4ac**
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* If D>0, both the roots will be real in nature.
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* If D==0, both the roots will be real and equal in nature.
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* If D<0, both the roots will be imaginary in nature (no real value of x will satisfy the equation)
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It can be easily observed that that the values of the roots are equal only when D==0 but the nature of roots is always the same for both roots.
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### Value of Roots
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Let the roots of **ax^2+bx+c=0** be p and q, then
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p= (-b + sqrt(D))/2a
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q= (-b - sqrt(D))/2a
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* The the equation has imaginary roots, they will always be found in conjugate pairs. For example, if you know that one of roots is 2+3i then you can directly determine the other root as 2-3i by just changing the sign in between the real and imaginary part of value. (This can be inferred from the formula of calculating roots' value.)
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